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Representations of a class of real $ B\sp *$-algebras as algebras of quaternion-valued functions


Author: S. H. Kulkarni
Journal: Proc. Amer. Math. Soc. 116 (1992), 61-66
MSC: Primary 46K05; Secondary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1992-1110546-4
MathSciNet review: 1110546
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Abstract: For a compact Hausdorff space $ X$, let $ C(X,{\mathbf{H}})$ denote the set of all quaternion-valued functions on $ X$. It is proved that if a real $ {B^*}$-algebra $ A$ satisfies the following conditions: (i) the spectrum of every selfadjoint element is contained in the real line and (ii) every element in $ A$ is normal, then $ A$ is isometrically $ *$-isomorphic to a closed $ *$-subalgebra of $ C(X,{\mathbf{H}})$ for some compact Hausdorff $ X$. In particular, a real $ {C^*}$-algebra in which every element is normal is isometrically $ *$-isomorphic to a closed $ *$-subalgebra of $ C(X,{\mathbf{H}})$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1110546-4
Article copyright: © Copyright 1992 American Mathematical Society

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